This project focuses on graph theoretical study of molecular evolution. Particular interests include developing graph theoretical methods, models, and algorithms for study multidomain proteins, protein-protein interactions and other biological networks.[unreadable] The project has the following subtopics:[unreadable] [unreadable] 1.Graph theoretical approaches to character based evolution. Here we consider the so called character overlap graph. The nodes of such graph are characters and there is an edge between two characters if they occur together in some organism. Understanding of properties of this graph provides insights into possible modes of evolution of the corresponding characters.[unreadable] [unreadable] 2.Recovering and representing the dynamics of protein-protein interaction networks. Our approach relies heavily on important results from graph theory; more specifically, the theory of two important graph families: chordal graphs and co-graphs. By using these two graph families in our modeling, we capture the dynamics of signaling pathways and overlapping structure of functional groups. [unreadable] [unreadable] 3.Predicting domain-domain interactions from protein-protein interaction network. We postulate that protein-protein interactions evolved in a parsimonious way. Consistently with the parsimony postulate, we identify the interacting domain as the smallest (weighted) set of pairs that is necessary to explain the protein interaction network.